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Modified: June 12, 2017.
 •  Continuous cluster categories II: Continuous cluster-tilted categories. (with Gordana Todorov)  
  Revising   Oct 11, 2014   
  Since CCCI has now appeared, we are working to finish part II. This paper starts with a review of the definitions and results of CCC I. This revised version has complete proofs of the main theorems, namely (1) the rational and continuous cluster-tilted categories are abelian and (2) they are isomorphic to categories of string modules over the Jacobian algebra of an infinite quiver with potential. Further revisions are needed to incorporate the precise descritption of the triangulation of the continuous categories given by the continuous Frobenius category.
 •  Periodic trees and pro-pictures. (with Kent Orr, Gordana Todorov, Jerzy Weyman)  
  Incomplete paper   Nov, 2014   
  This is a continuation of the paper Periodic trees and semi-invariants. The semi-invariant picture for an affine quiver is a inverse limit of pictures for an inverse system of finitely presented groups. The walls of the compartments separating the periodic functions for each period tree are separated by domains of these semi-invariants. This paper predates the picture groups series of papers. But we are redoing the definition of propicture group to match the (finite type) picture groups definition.
 •  Cluster morphism category as cubical category. (with Gordana Todorov)  
  Using a result of Speyer and Thomas and Lemma 1.3.1 from ``Picture groups and maximal green sequences'' we show that, for modulated quivers of finite type, the classifying space of the cluster morphism category is a cubical category whose geometric realization is a nonpositively curved cube complex for quivers of finite type. For tame infinite type, it is not npc and, therefore, not a K(pi,1). This gives another proof that the space X(Q) defined in the paper "Picture groups of finite type and cohomology in type A_n" is a K(pi,1).
 •  Picture groups and maximal green sequences. (with Gordana Todorov)  
  It is easy to see that there is a monomorphism from the set of maximal green sequences for an acyclic quiver into the set of positive expressions for the coxeter element of the corresponding picture group. This paper show that, in finite type, this mapping is a bijection. In the new version, the statement of Lemma 1.3.1 is modified.
See also my   Selected Papers

Home page: Kiyoshi Igusa Started: September   10, 2014  Last modified: (see top of page)
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